System and method for dynamic self-sensing of dielectric elastomer actuators

ABSTRACT

A system and method is provided for determining the capacitance between electrodes of an artificial muscle or dielectric elastomer actuator (DEA). The method comprises measuring the voltage difference between the electrodes of the DEA, the first derivative of that voltage with respect to time, and the total instantaneous current through the DEA, then calculating the capacitance of the DEA as the difference between the total instantaneous current through the DEA and the product of the voltage between the electrodes and an error term, divided by the first derivative of the voltage between the electrodes with respect to time. The capacitance may then be used to derive estimates of the leakage current, charge upon the DEA, and/or the physical state of the DEA, thereby implementing self-sensing to allow closed-loop feedback control of DEA actuation.

FIELD

This invention relates to the field of dielectric elastomer actuators.More particularly, the invention provides a system and method forobtaining feedback from a dielectric elastomer actuator (DEA) by way ofself-sensing.

BACKGROUND

Artificial muscles seek to replicate or mimic the versatility andcapability of natural skeletal muscles in exerting a mechanical force,but using electrical energy. Accordingly, an artificial muscle forms auseful electrical-mechanical transducer or actuator, exerting amechanical force through contraction and/or expansion.

Skeletal muscle is an amazingly versatile linear actuator. Traditionalactuation technologies such as piezoelectrics, electromagnetics, andshape memory alloys may be capable of out-performing skeletal muscle inspecific areas (e.g. speed, pressure, or energy density) but none arecapable of operating effectively in as wide a range of conditions asmuscle. By being highly compliant when not activated and only recruitingindividual muscle units as they are required, skeletal muscle has greatscope for optimizing efficiency for a wide range of loads and speeds.

Mimicking the well-rounded performance and characteristics of skeletalmuscle with an artificial actuator has not been possible withtraditional technologies. Electromagnetic motors for instance are heavyand rigid, and often must be coupled with a gearbox in order to achievea useful output, with each additional component or moving part addingits own inherent losses and complexity to the system. Piezoelectrics arecapable of high active speeds and pressures, but unlike muscle, they areextremely brittle and have very small output strains. Shape memoryalloys can produce high pressures and moderate strains, but are slow andsusceptible to fatigue loading. Dielectric Elastomer Actuators (DEAs),however, present a very promising alternative to these traditionaltechnologies. DEA performance in terms of strain, speed, and energydensity compare very favourably with those of skeletal muscle, andimportantly their low material density, compliant nature, and silentoperation capture many of the desirable physical properties of muscle.

Referring to FIGS. 1( a) and 1(b), a DEA generally referenced 10comprises a dielectric elastomer membrane 11 provided between compliantelectrodes 12. The dielectric elastomer membrane 11 is compressed byelectrostatic pressure when a high voltage is applied across theelectrodes 12 in the manner of a capacitor, causing planar expansion ofthe polymer from the uncompressed or contracted state as shown in FIG.1( a), to the compressed or expanded state illustrated in FIG. 1( b).

Natural muscle, however, is much more than just an actuator, as itprovides position feedback to the brain. Specialized cells within muscletissue provide feedback to the body's central nervous system and thisinformation is crucial to the coordination of muscle groups necessaryfor maintaining balance and posture. In an extreme case, that feedbackmay include a pain signal when there is a danger of overexertion causingdamage to the muscle or other parts of the body. Automatic reflexactions in response to this feedback can even occur without consciousthought, particularly in an attempt to prevent harm e.g. recoiling froma sharp object. Skeletal muscle is a key component in the distributedcontrol system that is the human body.

Because a DEA is constructed from a material which is resistant tocompression, it is possible to relate a change in capacitance to changesin the physical geometry of the DEA. “An adaptive control method fordielectric elastomer devices” (Todd A. Gisby, Emilio P. Calius, ShaneXie, and lain A. Anderson, Proc. SPIE, 2008), the contents of which areincorporated herein by way of reference, discloses the use ofself-sensing based upon the capacitance between electrodes to determinethe state of a DEA, thereby providing some feedback. Similar methods aredisclosed by “Control system design for a dielectric elastomer actuator:The sensory subsystem” (Toth, L. A. and A. A. Goldenberg, Proceedings ofSPIE, 2002) and “A self-sensing dielectric elastomer actuator” (Jung,K., K. J. Kim, and H. R. Choi, Sensors and Actuators A: Physical, 2008).

However, existing self-sensing methods are accurate only under certaincircumstances, such as when the DEA is stationary (i.e. not subject toany perturbations caused by external forces), the leakage current isnegligible, and/or for low actuation speeds, or are based uponassumptions which may not always hold true. In addition, the methods ofthe prior art may not be suitable for practical implementation in asystem designed for portable use. Accordingly, there is currently nosatisfactory method for accurately determining the capacitance or stateof a dielectric elastomer actuator, or providing feedback of movement ofan artificial muscle using self-sensing.

OBJECT OF THE INVENTION

It is therefore an object of the invention to provide improved feedbackon the state of an artificial muscle, or at least to provide the publicwith a useful choice.

Further objects of the invention will become apparent from the followingdescription.

SUMMARY OF INVENTION

According to a first aspect the invention may broadly be said to consistin a method of determining the capacitance between opposing electrodesof a dielectric elastomer actuator (DEA), the method comprising thesteps of:

-   -   measuring the voltage difference between the electrodes of the        DEA;    -   determining the first derivative of the voltage difference        between the electrodes of the DEA with respect to time;    -   measuring the total instantaneous current through the DEA; and    -   calculating the capacitance of the DEA as the difference between        the total instantaneous current through the DEA and the product        of the voltage between the electrodes of the DEA and an error        term, divided by the first derivative of the voltage between the        electrodes of the actuator with respect to time.

Preferably the error term is equal to the total instantaneous currentthrough the DEA divided by the voltage difference between the electrodesof the DEA when the first derivative of the voltage difference betweenthe electrodes of the DEA with respect to time is equal to zero.

Preferably the DEA is supplied by a pulse-width modulated (PWM) currentsource having a limited slew rate on either or both edges.

Preferably the slew rate of the current source is selected to enableaccurate detection of the zero-crossing of the first derivative of thevoltage difference between the electrodes of the DEA with respect totime, thereby minimising sensitivity to errors in the sample timing whensampling the total instantaneous current through the DEA.

Preferably the step of measuring the voltage difference between theelectrodes of the DEA comprises approximating the voltage using aresistor ladder.

Preferably the step of determining the first derivative of the voltagedifference between the electrodes of the DEA with respect to timecomprises approximating the first derivative using a differentiatorcircuit.

Alternatively, the step of determining the first derivative of thevoltage difference between the electrodes of the DEA with respect totime comprises approximating the first derivative using a finitedifference method on sequential measurements of the voltage differencebetween the electrodes of the DEA.

Preferably the step of measuring the total instantaneous current throughthe DEA comprises the step of measuring the voltage difference across aknown series resistance.

According to a second aspect the invention may broadly be said toconsist in a method of determining the leakage current between opposingelectrodes of a dielectric elastomer actuator (DEA), the methodcomprising the steps of:

-   -   determining the capacitance between the electrodes according to        the method of the first aspect of the invention;    -   calculating the first derivative of the capacitance with respect        to time; and    -   calculating the leakage current as the difference between the        total instantaneous current through the DEA and the product of        the voltage difference across the DEA and the first derivative        of the capacitance with respect to time, at a point in time when        the first derivative of the voltage difference across the DEA        with respect to time is substantially equal to zero.

According to a third aspect the invention may broadly be said to consistin a method of determining the state of a dielectric elastomer actuator(DEA), the method comprising the steps of:

-   -   determining the capacitance between opposing electrodes of the        actuator according to the method of the first aspect of the        invention; and    -   determining the state of the DEA corresponding with the        capacitance.

Preferably the step of determining the state of the DEA comprises thesteps of:

-   -   a) determining the ratio of the instantaneous capacitance of the        DEA to its initial capacitance;    -   b) relating the ratio of step a) to the ratio between the        instantaneous planar area of the DEA to its initial planar area;        and    -   c) determining the displacement of the DEA from the relationship        determined in step b).

According to a fourth aspect the invention may broadly be said toconsist in a method of controlling a dielectric elastomer actuator (DEA)comprising the steps of:

-   -   determining the state of the DEA according to the method of the        third aspect of the invention; and    -   controlling the charge on the electrodes of the DEA according to        the difference between the determined state and a desired state        of the DEA.

Preferably the method further comprises the steps of:

-   -   determining the leakage current according to the second aspect        of the invention; and    -   limiting the charge upon the electrodes when the leakage current        exceeds a predefined threshold.

According to a fifth aspect the invention may broadly be said to consistin a system for determining the capacitance between opposing electrodesof a dielectric elastomer actuator (DEA), the system comprising:

-   -   a DEA monitoring circuit to obtain feedback information        regarding the electrical behaviour of DEA; and    -   computation means adapted to calculate the capacitance of the        DEA as the difference between the total instantaneous current        through the DEA and the product of the voltage difference across        the DEA and an error term, divided by the first derivative of        the voltage difference across the actuator with respect to time.

Preferably the DEA monitoring circuit comprises:

-   -   a pulse-width modulated current source;    -   a dielectric elastomer actuator (DEA);    -   means for measuring the voltage difference across the DEA;    -   means for measuring the first derivative of the voltage        difference across the DEA with respect to time; and    -   means for measuring the total instantaneous current through the        DEA.

Preferably the means for measuring the voltage difference comprises aresistor ladder, the means for measuring the first derivative of thevoltage difference comprises a differentiator circuit, and the means formeasuring the current through the DEA comprises a series resistor.

Preferably the error term is equal to the total instantaneous currentthrough the DEA divided by the voltage difference across the DEA whenthe first derivative of the voltage difference across the DEA withrespect to time is equal to zero.

Preferably the computation means is further adapted to determine thestate of the DEA corresponding with the calculated capacitance.

Preferably the computation means is further adapted to control the stateof the DEA by way of controlling the current source.

According to a sixth aspect the invention may broadly be said to consistin a method of modelling or simulating a dielectric elastomer actuator(DEA), the method comprising the step of representing the DEA as anideal capacitance augmented with an equivalent parallel resistance toaccount for leakage current, and an equivalent series resistance toaccount for the resistance of the electrodes.

In a further aspect the invention may broadly be said to consist in amethod of determining the state of a DEA, the method comprising thesteps of:

-   -   a) determining the ratio between the total instantaneous current        through the DEA and the voltage difference across the DEA at a        time when the change in voltage difference across the DEA is        zero or close to zero;    -   b) multiplying the ratio of step a) with the voltage difference        across the DEA when the change in voltage is non-zero;    -   c) subtracting the product of step b) from the total        instantaneous current through the DEA;    -   d) dividing the sum of step c) by the instantaneous change in        voltage difference across the DEA with respect to time; and    -   e) using the product of step d) to determine the state of the        DEA.

Further aspects of the invention, which should be considered in all itsnovel aspects, will become apparent from the following description.

DRAWING DESCRIPTION

A number of embodiments of the invention will now be described by way ofexample with reference to the drawings in which:

FIG. 1 shows a dielectric elastomer actuator (DEA) according to theprior art, in (a) uncompressed, and (b) compressed states;

FIG. 2 shows a graph of the error between the predicted and actualvalues of capacitance during transient states of the actuator using aquasi-static method of determining capacitance according to the priorart;

FIG. 3 is a schematic of an example DEA monitoring circuit according tothe present invention;

FIG. 4 illustrates the application of Kirchoff's current law to thecircuit of FIG. 3;

FIG. 5 shows the effect the slew rate of the current source has on therate of change of i_(Rs);

FIG. 6 shows the leakage current versus electric field for an exampleDEA;

FIG. 7 is a graph of actual and predicted capacitance versus time forthe quasi-static method of the prior art;

FIG. 8 is a graph of actual and predicted capacitance versus time forthe dynamic method of the present invention;

FIG. 9 shows the leakage current versus time over the course of thesimulation;

FIG. 10 shows the first derivative of capacitance of the DEA withrespect to time versus time over the course of the simulation; and

FIG. 11 is a block diagram of an example system using the feedbackmethod of the present invention.

DETAILED DESCRIPTION OF THE DRAWINGS

Throughout the description like reference numerals will be used to referto like features in different embodiments.

Structurally, dielectric elastomer actuators (DEAs) resemble a compliantcapacitor consisting of a resilient soft polymer membrane dielectric 11with compliant electrodes 12 on substantially opposing sides of themembrane. The charge accumulated on the electrodes 12 after a voltage isapplied gives rise to electrostatic forces that generate deformation inthe DEA 10. The net result of the interaction between the positive andnegative charges and the mechanical properties of the DEA produces acoupled decrease in dielectric membrane 11 thickness and increase inmembrane planar area, as shown in FIG. 1( b). When the charge isremoved, the elastic energy stored in the dielectric returns it to itsoriginal shape as shown in FIG. 1( a).

The charge-generated surface pressure, or Maxwell stress, isproportional to the square of the electric field within the dielectricand can be described by Equation 1. P is the pressure, ∈_(r) is therelative permittivity of the dielectric material, ∈₀ is the permittivityof free space (8.854×10⁻¹² F/m), V is the voltage, and d is thedielectric membrane thickness in meters.

$\begin{matrix}{P = {ɛ_{r}{ɛ_{0}\left( \frac{V}{d} \right)}^{2}}} & (1)\end{matrix}$

The present invention seeks to provide similar feedback functionality asthat of natural muscle by integrating a sensing ability. Accordingly, afeedback system according to the present invention provides feedback onthe state of a dielectric elastomer actuator (DEA) 10, giving anindication of the extent to which the membrane 11 is compressed, and/orthe corresponding planar expansion thereof.

Sensing and feedback in actuators according to the prior art havetypically required the incorporation of sensors such as optical sensors,strain gauges, micro-switches and the like which are external to thecomponents of the actuator itself, creating an actuator/sensor hybrid.While the benefit of feedback allows improved control of the actuator,the sensors increase the component count and complexity of the actuator.An important property of DEAs is the potential for self-sensing, i.e.sensing an electrical property of the actuator itself. In particular,the state of a DEA can be determined by sensing the capacitance betweenthe electrodes 12.

A self-sensing artificial muscle has the potential to be more compactand simpler to construct than actuators incorporating external sensorsowing to reduced component count, and may also provide a built-in safetymechanism whereby the health of the DEA can be monitored, as will bedescribed in further detail below. However, due to the high voltagesapplied to the electrodes which are necessary to actuate a DEA,implementing capacitive self-sensing is not as simple as applying thecapacitive sensing techniques commonly applied in other fields.

DEAs represent a highly coupled electro-mechanical system. Where theyare coupled to a “real world” load susceptible to non-deterministicdisturbances, controlling them with any degree of accuracy becomes aninteresting challenge. There is a complex exchange of electrical andmechanical energy; a high voltage input must be applied to achievemechanical actuation, yet the mechanical response of the DEA (whetherdue to the electrical input energy or an external disturbance) affectsthe necessary input to achieve the desired output state. Feedback isnecessary to achieve accurate control.

Self-sensing can be implemented by measuring the capacitance of the DEA.Owing to the resilience or tendency of the dielectric membrane 11 toreturn to its original uncompressed state, it is possible to relate thecapacitance of a DEA 10 to its state of deformation. Measuring thecapacitance of a DEA 10 is important because with the instantaneouscapacitance and voltage, the instantaneous electrostatic charge on theDEA can be calculated and therefore controlled. Controlling chargeresults in a stable system where the electrostatic response of the DEAwill act to reject external physical disturbances. For instance if theDEA deforms such that the thickness is reduced, the capacitance willincrease. If a constant charge is maintained, the voltage differenceacross the DEA must drop and the electric field will therefore decrease,reducing the effective electrostatic pressure. The elastic energy in thedielectric membrane 11 will also increase, and both of these effectswill create a net effect counteracting the influence of the disturbance.In contrast, if the charge did not remain constant i.e. if the voltagewas held constant, an increase in the capacitance of the DEA will meanmore charge would flow onto the DEA 10. This extra charge will serve toincrease the electric field and subsequently the electrostatic pressure.If the electric field is allowed to grow too large the DEA will undergodielectric breakdown. When this happens the charge on the DEA will berapidly discharged through the thickness of the membrane 11, generatingsignificant heat and often resulting in catastrophic failure of the DEA10. However, while it may for this reason generally be preferable tomaintain a constant charge on the DEA electrodes, the charge mayalternatively be allowed to vary without departing from the scope of thepresent invention.

Prior examples of self-sensing include the systems presented by Toth andGoldenberg, and Jung et al., as referenced above. Both have createdsystems whereby the capacitance, and hence displacement, can be measuredwhile the DEA is being actuated. In these integrated sensor/actuatorsystems the DEA is treated as the capacitive component of an RC filter,with the dynamic response of the DEA circuit to a low amplitude, highfrequency sensory signal overlaid on the high amplitude, low frequencyactuation voltage used to calculate capacitance.

According to the present invention, a slew-rate controlled PWM signal ispreferably used to create an input current waveform that creates both aDC offset for actuation and a dynamic high frequency excitation used formeasuring capacitance. This approach builds upon the self-sensingtechnique for quasi-static DEA systems disclosed by Gisby et al. (ToddA. Gisby, Emilio P. Calius, Shane Xie, and lain A. Anderson, “Anadaptive control method for dielectric elastomer devices”, Proc. SPIE,2008), the contents of which are incorporated herein by reference. Morespecifically, the preferred capacitive self-sensing technique takes intoaccount the effects of leakage current and a non-zero first derivativeof the DEA capacitance with respect to time, both of which were ignoredin the self-sensing techniques of the prior art. Ignoring these effectsresults in a significant error in the predicted value of the capacitanceduring transient states of the actuator, as shown in FIG. 2.Accordingly, an electrical representation of a “real” DEA has beendeveloped to determine how to approximate the influence of electricalbehaviours within the DEA (that cannot be measured directly) usingelectrical parameters that can be measured.

For reasons outlined above a system using the instantaneous charge onthe DEA as the primary control variable is preferred. However, chargecannot be measured directly therefore DEA capacitance and voltage mustbe known in order to calculate charge according to Equation 2.

Q=CV  (2)

Voltage can easily be measured using a resistance ladder network, forexample, but in order to calculate capacitance it is necessary to lookat the derivative of Equation 2, which for an ideal capacitor is givenin Equation (3).

$\begin{matrix}{i = {{C\frac{V}{t}} + {\frac{C}{t}V}}} & (3)\end{matrix}$

A DEA, however, is not an ideal capacitor. In reality the DEA isrepresented by an ideal capacitance (C_(dea)) augmented with anequivalent parallel resistance (R_(epr)) in parallel to account forleakage current and an equivalent series resistance (R_(esr)) to accountfor the resistance of the electrodes. The nature of the DEA means thatit is impossible to separate the current flowing through the capacitiveelement of the DEA and the leakage current without assuming the DEA isbehaving in some predetermined manner. It is necessary therefore to findsome way of relating the effects of these currents to electricalcharacteristics that can be measured. The example DEA monitoring circuitillustrated in FIG. 3 has been designed to obtain feedback informationregarding the electrical behaviour of the circuit, wherein the DEA 10 isrepresented by the capacitor C_(dea) (representing the capacitancebetween the electrodes 12) and resistors R_(esr) and R_(epr) within thedotted-line box. C_(dea), R_(epr), and R_(esr) will vary depending onthe electromechanical loading of the DEA. R_(p1) and R_(p2) represent avoltage divider ladder used to directly measure the voltage differenceacross the DEA. C_(hpf) and R_(hpf) represent a simple high pass filterthat acts as a differentiator circuit and can be used to measure therate of change of voltage across the DEA. The circuit has been designedto be driven by a low power, high voltage DC-DC converter coupled with ahigh voltage opto-coupler (such as the OC100HG, available from VoltageMultipliers, Inc.) or other suitable switch. The properties of thesecomponents combined can effectively be modelled as a current source.

Kirchoff's Current Law states the sum of all currents entering andexiting a junction in a circuit must be zero. Using this relationshipthe current through resistor R_(s) (i_(Rs), effectively the totalinstantaneous current through the actuator) can be defined by Equation4, where i_(dea) and i_(epr) are the currents through C_(dea) andR_(epr) respectively.

i _(Rs) =i _(dea) +i _(epr)  (4)

Equation 4 can be expanded to incorporate the full expression fori_(dea) (from Equation 3) and i_(epr), to become Equation 5, whereV_(dea) is the voltage across the capacitor C_(dea).

$\begin{matrix}{i_{Rs} = {{C_{dea}\frac{V_{dea}}{t}} + {\frac{C_{dea}}{t}V_{dea}} + \frac{V_{dea}}{R_{epr}}}} & (5)\end{matrix}$

Equation 5 states that the overall current through the DEA is a functionof the capacitance and the voltage, the first derivatives of both ofthese parameters, and the leakage current. If leakage current throughthe DEA and dC_(dea)/dt are sufficiently small (i.e. if the actuator ismoving slowly/stationary) the second and third terms of Equation 5 canbe ignored and calculating capacitance becomes straightforward usingEquation 6 as described by Gisby et al.

$\begin{matrix}{C_{dea} = \frac{i_{Rs}}{\left( \frac{V_{dea}}{t} \right)}} & (6)\end{matrix}$

The limitation of this approach is that ignoring the latter two terms ofEquation 5 introduces an error that is proportional to the sum of thespeed of actuation and the leakage current. Leakage current, for thecommon DEA membrane material VHB4905 at least, begins to degrade theaccuracy of the calculation for electric fields greater thanapproximately 80-100 MV/m as shown in FIG. 6. A more advanced approachis necessary to calculate capacitance where these simplifyingassumptions do not hold.

Examining Equation 5 it is clear five parameters must be known in orderto calculate C_(dea): i_(Rs), dV_(dea)/dt, dC_(dea)/dt, V_(dea), andR_(epr). V_(dea) and dV_(dea)/dt can be approximated using the resistorladder and differentiator circuit respectively, and i_(Rs) can bemeasured directly. Measuring i_(epr) and dC_(dea)/dt for all conditionshowever is impossible due to the closed loop formed by C_(dea) andR_(epr) internal to the DEA. For this reason it is necessary to lump thecontribution of i_(epr) and dC_(dea)/dt to the calculation of C_(dea)into a combined term. Further simplification however is still requiredif C_(dea) is to be calculated. This can be done by measuring i_(Rs)when dV_(dea)/dt is equal to zero, thereby eliminating the first term onthe right hand side of Equation 5 to arrive at Equation 7.

$\begin{matrix}{i_{Rs} = {V_{dea}\left( {\frac{C_{dea}}{t} + \frac{1}{R_{epr}}} \right)}} & (7)\end{matrix}$

The influence of dC_(dea)/dt and R_(epr) are then combined intoK_(error) as given by Equation 8.

$\begin{matrix}{K_{error} = \frac{i_{Rs}}{V_{dea}}} & (8)\end{matrix}$

Now, while their relative sizes of dC_(dea)/dt and R_(epr) are unknown,their combined effect is. By assuming dC_(dea)/dt and R_(epr) remainconstant around the point where dV_(dea)/dt is zero and substitutingEquation 8 back into Equation 5 and evaluating it for any point wheredV_(dea)/dt is non-zero, the final capacitance can be calculated usingEquation 9.

$\begin{matrix}{C_{dea} = \frac{\left( {i_{Rs} - {V_{dea}K_{erorr}}} \right)}{\left( \frac{V_{dea}}{t} \right)}} & (9)\end{matrix}$

Broadly speaking, therefore, the method involves calculating anelectrical characteristic (capacitance) of the circuit from othercharacteristics (i_(Rs), dV_(dea)/dt, dC_(dea)/dt, V_(dea), and R_(epr))which can be measured directly.

Once the DEA capacitance C_(dea) is determined, this value can be usedto estimate or determine the position or state of the actuatorcorresponding therewith. By relating the instantaneous capacitance ofthe DEA to its initial capacitance (i.e. the capacitance between theelectrodes when the DEA is at rest or equilibrium), the ratio (λ_(A)) ofthe instantaneous planar area of the DEA to its initial planar area, andtherefore the displacement of the DEA, can be determined using therelationship of Equation 10.

$\begin{matrix}{\lambda_{A} = \sqrt{\frac{C_{dea}}{C_{initial}}}} & (10)\end{matrix}$

Accordingly, the method provides more accurate feedback for the controlof artificial muscles, whereby a computation means calculating thecapacitance may be further adapted to control the artificial muscle byway of controlling the pulse width or duty cycle of a current sourcesupplying the DEA such that the actual (determined) position or state ofthe actuator substantially matches a required state, thereby forming aclosed feedback loop for more precise control of a mechanical systemwith which the artificial muscle may be coupled.

Each time dV_(dea)/dt is equal to zero it is therefore possible tocalculate C_(dea). If the existing and previous estimations ofcapacitance are known (C_(dea) and C_(dea(previous)) respectively), andthe time between these estimations (t) is also known, a backwardsdifference approximation can be used to estimate dC_(dea)/dt. Equation 5can then be rearranged and used to back calculate the leakage current(i_(epr)) according to Equation 11.

$\begin{matrix}{\frac{V_{dea}}{R_{epr}} = {i_{epr} = {i_{Rs} - {C_{dea}\frac{V_{dea}}{t}} - {\frac{\left( {C_{dea} - C_{{dea}{({previous})}}} \right)}{t}V_{dea}}}}} & (11)\end{matrix}$

Monitoring the leakage current can enable detection of the precursors todielectric breakdown and failure of the DEA. It can therefore be used tostop overcharging of the membrane which leads to breakdown. For example,the charge on the DEA may be limited when the leakage current exceeds apredefined threshold. Detecting and preventing breakdown not onlyincreases device reliability, but also enables the DEA to be drivencloser to its performance limits with greater confidence.

A numerical model of the circuit presented in FIG. 3 can be created inThe MathWorks, Inc.'s MATLAB®, for example, to test the effectiveness ofthe new self-sensing approach compared with the simplified quasi-staticexpression described by Equation 6. The model simulates behaviour of theDEA system for conditions where the input control signal is heldconstant while the capacitance changes i.e. a situation analogous to theDEA being deformed by an external load. To do this the model takesarbitrary functions for input current, DEA capacitance, and DEA leakagecurrent as inputs, and incrementally solves for the current through eachbranch of the circuit at each time step. The initial electricalcharacteristics of the components comprising the DEA have been modelledon experimental data obtained from a simple 24 mm diameter expanding dotactuator made from a VHB4905 membrane stretched equibiaxially to 16times its original area and bonded to a rigid circular plastic frame.This actuator has an initial dielectric membrane thickness of 31.25 μmand a rest capacitance of approximately 500 pF.

FIG. 4 shows the schematic for the DEA 10 electrical subsystem that isthe basis of the numerical model. Again using Kirchoff's Current Law,the current entering/exiting the upper junction 40 in FIG. 4 isexpressed by Equation 12, where i_(in) is the user defined inputcurrent, i_(Rp) is the current through the resistor ladder used tomeasure the voltage across the DEA, i_(dea) is the charging/dischargingcurrent flowing through the DEA, i_(epr) is the leakage current throughthe thickness of the membrane, and i_(hpf) is the current through thehigh pass filter.

i _(in) =i _(Rp) +i _(dea) +i _(epr) +i _(hpf)  (12)

The instantaneous current through the DEA (i_(dea)) is evaluated forevery time step and integrated to calculate the charge stored on theDEA. After each iteration of the model, the instantaneous DEAcapacitance at the next time step is interpolated from the capacitanceinput function, and the voltage across the DEA for the next iteration isfound by dividing the new charge by the new capacitance.

The current i_(Rp) is easily evaluated as the sum of the DEA voltage andthe voltage across the known series resistance divided by the totalresistance of the external resistor ladder. The leakage current throughthe membrane (i_(epr)) is modelled as an electric field dependentparameter. A relationship between electric field and leakage current wasfound by fitting an exponential curve to electric field versus leakagecurrent data obtained experimental using the VHB4905 test actuator. Themathematical representation of this curve was then used to calculate theinstantaneous i_(epr) for each time step in the model simulation. Thefunction for the leakage current can be defined completely arbitrarily,it is simply to add a degree of realism that it is based uponexperimental data for a real DEA. Both the quasi-static and dynamicmethods for calculating capacitance rely solely on parameters that canbe measured directly i.e. i_(Rp), i_(Rs), and i_(hpf).

Before i_(epr) can be calculated it is necessary to calculate theelectric field in the DEA. This requires knowledge of the voltage acrossthe DEA and the instantaneous membrane thickness. The voltage can beeasily calculated based on previous iterations of the model, and,assuming uniform deformation of the DEA, the instantaneous membranethickness can be calculated based on a function of the ratio of theinitial membrane thickness initial capacitance and the instantaneouscapacitance as given by Equation 13.

$\begin{matrix}{d = {\left( \sqrt{\frac{C_{{dea}{({rest})}}}{C_{dea}}} \right)d_{rest}}} & (13)\end{matrix}$

With i_(in), i_(Rp), and i_(epr) now calculated, the remaining currentnot accounted for is shared between i_(hpf) and i_(dea). The fixedcapacitance of the passive high pass filter means i_(hpf) is solelyproportional to the rate of change of the voltage across the capacitor.The shared positive electrode between the capacitive element of the DEAand the high pass filter means the rate of change of the voltage acrossboth can be approximated as being the same (Equation 14).

$\begin{matrix}{\frac{V_{dea}}{t} = \frac{i_{hpf}}{C_{hpf}}} & (14)\end{matrix}$

Substituting Equation 14 into Equation 5 gives Equation 15.

$\begin{matrix}{i_{Rs} = {{C_{dea}\frac{i_{hpf}}{C_{hpf}}} + {\frac{C_{dea}}{t}V_{dea}} + \frac{V_{dea}}{R_{epr}}}} & (15)\end{matrix}$

Substituting Equation 15 into Equation 12 gives Equation 16.

$\begin{matrix}{i_{hpf} = \frac{\left( {i_{in} - i_{Rp} - {i_{{epr} -}\frac{C_{dea}}{t}V_{dea}}} \right)}{\left( {1 + \frac{C_{dea}}{C_{hpf}}} \right)}} & (16)\end{matrix}$

Once i_(hpf) is known, i_(dea) can be evaluated algebraically usingEquation 12.

It is necessary to measure i_(rs) at the point where dV_(dea)/dt isequal to zero in order to calculate K_(error). For a periodic inputsignal, this occurs during the periods where the input current is eitherramping up or ramping down. For simplicity the model employed herefocuses on the period for which the input current is ramping down,however it could equally be applied to when the input current is rampingup or indeed both ramping up and down.

If the input current is controlled using a pure PWM signal it willinherently have a very high slew rate, which means capturing data at thezero-crossing or the exact point where dV_(dea)/dt equals zero requiresprohibitively fast data acquisition capabilities. By introducing alimited slew rate to the input signal, the transition of dV_(dea)/dtprogressing from being positive to being negative is spread over alonger time period. This reduces the sensitivity of sampling i_(rs) toerrors in the sample timing. FIG. 5 shows quantitatively the effect slewrate has on the rate of change of i_(rs) (note the slew rate has onlybeen applied to the falling edge of the input signal). For the numericalmodel, the input current was limited to a maximum of 100 uA, and amaximum slew rate of 200 mA/s was imposed on the falling edge of theinput current waveform. A suitable slew rate is dependent on theactuator and the system.

Leakage current data versus electric field for a VHB4905 expanding dotactuator stretched equibiaxially to 16 times its original area ispresented in FIG. 6. Also shown in FIG. 6 is an exponential curve thathas been fitted to the raw data to provide a mathematical relationshipbetween electric field and leakage current for use in the numericalmodel. The fitted curve used in the model has the parameters of Equation17.

$\begin{matrix}{i_{epr} = {2.458 \times 10^{- 6}\left( \frac{E_{field}}{180 \times 10^{6}} \right)^{63.0625}}} & (17)\end{matrix}$

An arbitrary input capacitance waveform was created with featurescorresponding to 5 conditions: a constant capacitance; a slow increasein capacitance; a fast increase in capacitance; a fast decrease incapacitance; and a slow decrease in capacitance. The input currentwaveform was fixed as a 500 Hz PWM signal with a maximum value of 100 uAand a limiting slew-rate of rate of 200 mA/s was applied to the fallingedge of the PWM signal. Leakage current was calculated based on theinstantaneous electric field according to the relationship described inEquation 17. The simulation used a step size of 2 μs.

FIG. 7 displays the actual DEA capacitance compared to that predicted bythe quasi-static self-sensing method based on Equation 6 in which theeffects of dC_(dea)/dt and leakage current are ignored. FIG. 8 comparesthe actual DEA capacitance with that predicted by the new dynamicself-sensing method described by Equation 9 in which the influence ofdC_(dea)/dt and leakage current are approximated using K_(error). FIGS.10 and 11 show the leakage current and the magnitude of dC_(dea)/dtrespectively over the course of the simulation.

As can be seen from FIGS. 7 and 8, the new dynamic self-sensing methodoffers a significant advantage in the accuracy between predicted andactual DEA capacitance compared to the previously presented quasi-staticself-sensing method based on PWM. The accuracy of the quasi-staticmethod is extremely sensitive to the rate of change of the DEA'scapacitance. The magnitude of the error can clearly be seen in to belarge where the absolute value of dC_(dea)/dt is significant. For thisreason the quasi-static approach to calculating capacitance isunsuitable for situations where high speed actuation, whether it is theresult of a change in input signal or an external disturbance, is apossibility.

The quasi-static method also lacks any inherent compensation for theeffects of leakage current through the dielectric membrane. In FIG. 8the impact of leakage current manifests itself in the steady state errorobserved between the true capacitance and the predicted capacitance onthe flat region of the capacitance curve from 0.24 seconds to 0.29seconds. Leakage current increases exponentially with electric field andbecomes significant for electric fields greater than approximately 100MV/m. Anecdotal evidence suggests that for VHB4905 actuators operatingabove this electric field threshold typically will reduce the lifetimeof the DEA considerably. Instead a maximum operating electric field ofapproximately 60-80 MV/m produces longer-lasting devices. It must bepointed out, however, that better feedback from the DEA may extend thesafe range of operation into the region where leakage current issignificant. It is also important to consider that different dielectricmembrane materials may have different leakage current characteristicsthan those observed for VHB4905. It is important therefore to accountfor this current where possible.

In contrast to the quasi-static method, the dynamic method isconsiderably more accurate for determining the DEA capacitance over theentire range of input conditions. The dynamic capacitance does lagbehind the true capacitance by a small amount; however this lag isindependent of the rate of change of capacitance and can easily becompensated for by the control system. In every other respect thepredicted capacitance shows very good agreement with the actualcapacitance. In particular there is no steady state error between thetrue capacitance and the dynamic prediction for the period from 0.24seconds to 0.29 seconds, where leakage current is at its maximum. Thedynamic method, in accounting for errors due to dC_(dea)/dt and leakagecurrent, effectively addresses the observed shortfalls of thequasi-static method. Being able to back-calculate for the leakagecurrent is also an important and valuable step as it has been suggestedthat leakage current can enable detection of the precursors todielectric breakdown. Detecting and preventing breakdown would not onlyincrease device reliability, but also enable the DEA to be driven closerto its performance limits with greater confidence.

The method of calculating capacitance requires dV_(dea)/dt to be equalto zero in order to calculate the lumped error term K_(error), asdescribed above. At high speeds the dC_(dea)/dt component may be sogreat that the derivative of the voltage across the DEA does not crosszero at any point during the PWM cycle. In this situation it is notpossible to calculate the K_(error) and therefore it is not possible toaccurately compensate for this term. However, dV_(dea)/dt may still beused to determine if the DEA is converting mechanical energy intoelectrical energy. In this situation the control system could switch toa generator mode and use the electrical energy created to recharge itspower supply, for example.

The properties, performance and functionality of Dielectric ElastomerActuators make them ideal candidates for artificial muscles. Trueartificial muscles however must be capable of acting as both sensor andan actuator at the same time. A numerical model of the electricalbehaviour of a DEA has been created that incorporates the “non-ideal”characteristics of leakage current and the effects of a variablecapacitance. This model has then been used to compare two methods forcalculating the capacitance of a DEA: one simple quasi-static method inwhich the non-ideal elements are ignored; and another more advanceddynamic method that combines the effects of the non-ideal elements intoa lumped parameter. The numerical model clearly shows the error betweenthe actual capacitance and the predicted capacitance for thequasi-static method is highly sensitive to the rate of change ofcapacitance and only provides accurate measurements when the absolutevalue of dC_(dea)/dt is small. The more dynamic method however addressesthe shortfalls of the quasi-static method and has been shown to berobust to changes in capacitance and variable leakage current. It alsoenables leakage current to be calculated which may prove a valuable stepin detecting dielectric breakdown before it happens and subsequentlygreatly improves the reliability of DEA devices.

The method described herein above may be implemented in a system fordetermining the capacitance, leakage current, and/or state/position of adielectric elastomer (or a lever coupled therewith, for example)including a computation means. Where the method of calculatingcapacitance is intended to form a closed feedback loop for controllingthe DEA, the computation means is preferably further adapted to controlthe DEA by varying the pulse-width of a current supply. Capacitiveself-sensing in a DEA has been found to be robust to externally induceddeformations of the DEA. A single power supply may be used to powermultiple DEAs.

An example of a system including a feedback loop according to thepresent invention is shown in FIG. 11. In this example, the systemcomprises a DEA 10 mechanically coupled with a load 110 for controlleddisplacement thereof. The DEA is supplied by a pulse-width modulated(PWM) high voltage current source 111. The PWM signal is supplied andcontrolled by the computation means 112 on the basis of feedback fromthe feedback sensors 113, which together with the DEA 10 form the DEAmonitoring circuit described previously. The feedback sensors 113comprise means for providing an indication or measurement of the voltagedifference between electrodes of the DEA, the first derivative of thevoltage difference between the electrodes of the DEA with respect totime, and the total instantaneous current through the DEA. Those meansin the preferred embodiment of the invention comprise a resistor ladder,a differentiator circuit, and a series resistor (across which thepotential difference can be measured to derive the currenttherethrough), respectively, although it is to be appreciated that anyother suitable measurement or sensing means may alternatively be usedwithout departing from the scope of the invention, and the selection andimplementation of such sensing means is within the ambit of skills ofany person knowledgeable in the area of electronic circuit design.

The current source 111, computation means 112, and feedback sensors 113can each be supplied by a low voltage power supply 114.

The computation means 112 is programmed or otherwise adapted tocalculate the capacitance of the DEA from other characteristics (i_(Rs),dV_(dea)/dt, dC_(dea)/dt, V_(dea), and R_(epr)) measured by the feedbacksensors, and to relate this electrical characteristic to the physicalstate of the DEA 10. Any mismatch between the desired and calculatedactual state may be corrected by adjusting the duty cycle of the PWMcontrol signal to supply more or less charge to the DEA according to thedifference, using proportional, integral, or differential control, orany combination thereof, for example. Any suitable control scheme may beused without departing from the scope of the invention.

According to a preferred lightweight portable embodiment, the system isbattery powered and the computation means comprises a microcontroller.Other computer system configurations can also be employed to perform themethod of this invention, and to the extent that a particular systemconfiguration is capable of performing the method of this invention, itis equivalent to the representative digital computer system described,and within the scope and spirit of this invention. Once they areprogrammed to perform particular functions pursuant to instructions fromprogram software that implements the method of this invention, suchdigital computer systems in effect become special-purpose computersparticular to the method of this invention. The techniques necessary forthis are well-known to those skilled in the art of computer systems.

A microcontroller or other computation means 112 adapted to control aDEA using a method of self-sensing feedback as described herein abovemay be provided to control each artificial muscle in a mechanicalsystem. The computation means 112 or microcontrollers, according to suchan embodiment, may therefore provide distributed control of the musclesin the system. Alternatively, or additionally, each microcontroller inthe system may be communicatively coupled with a central control meansacting to coordinate movement of the muscles in the system bycommunicating the movement required of each muscle to the respectivemicrocontroller, and each microcontroller acts to control the respectivemuscle using the self-sensing dynamic feedback method. In such a system,each microcontroller can actuate the muscle independently based in parton information or knowledge of the muscle, such as breakdown strength ofthe dielectric material or its frequency dependent properties inaddition to the calculated leakage current and/or capacitance. Themicrocontroller can therefore monitor properties of the respectiveartificial muscle and automatically shut it down or limit the charge ifthe electric field is getting too high or the leakage current too great,preferably independently of any central control means. Such a circuitwould be acting locally, to avoid damage without involving a highersystem control, as in an autonomous nervous system. Thus the muscleoutput can be optimized, reliability improved, and damage to the musclecan be avoided as the system will know its limits. Alternatively, asingle computation means 112 may be used to simultaneously control twoor more DEAs or artificial muscles as described herein.

From the foregoing it will be seen that an improved method ofdetermining the capacitance, and therefore state, of a dielectricelastomer actuator is provided, which takes into account the both thespeed of actuation and the leakage current. Accordingly, the methodprovides an accurate estimate of the dynamic capacitance and state of aDEA which is more suitable for providing feedback and therefore moreprecise control of artificial muscles. In particular, the methods andsystems according to the invention provide a substantially accurateinstantaneous indication of the continuously varying or dynamiccapacitance of a DEA, whether through actuation or externalperturbations, which may be used to more accurately determine the stateof the DEA and provide more accurate and robust control thereof viaclosed-loop feedback.

Additionally, the system may be practically implemented in real-lifeapplications, including those requiring portability and/or mobility. Inmobile robotics applications, for example, it is important that thebenefits of a soft, lightweight muscle-like actuator are not outweighedby the size and/or weight of bulky driving and support circuitry suchthat it becomes inefficient to use DEA. The present invention provides arelatively compact and simple system for controlling a DEA actuator,providing feedback on the actuator state without the need for externalposition sensors. For example, for a prosthetic hand using one or moreDEA actuators as artificial muscles to be truly capable of replicatingthe functionality of a human hand requires accurate control and lowmass, and it must not be tethered to a fixed installation (e.g. a mainspower supply). Feedback is necessary to accurately control theartificial muscles in the prosthetic hand in an environment where a widerange of external loading conditions and/or disturbances are to beexpected. Low mass is required to reduce the inertia of the device onthe end of the wearer's arm.

In another example application, the system and/or method may be used toimplement mechano-sensitivity allowing de-centralised control andcoordination of a plurality of DEA actuators in an array, wherein theactuators themselves are each adapted to propagate an actuation signalby triggering actuation upon detecting deformation caused by contactfrom an adjacent actuator or a load. Such systems are disclosed only asnon-limiting examples of applications of the present invention, and itis to be appreciated that the system and/or method may be adapted for awide range of other applications without departing from the scope of theinvention.

Unless the context clearly requires otherwise, throughout thedescription, the words “comprise”, “comprising”, and the like, are to beconstrued in an inclusive sense as opposed to an exclusive or exhaustivesense, that is to say, in the sense of “including, but not limited to”.

Although this invention has been described by way of example and withreference to possible embodiments thereof, it is to be understood thatmodifications or improvements may be made thereto without departing fromthe scope of the invention. Furthermore, where reference has been madeto specific components or integers of the invention having knownequivalents, then such equivalents are herein incorporated as ifindividually set forth.

Any discussion of the prior art throughout the specification should inno way be considered as an admission that such prior art is widely knownor forms part of common general knowledge in the field.

1. A system for determining the capacitance between opposing electrodesof a dielectric elastomer actuator (DEA), the system comprising: a DEAmonitoring circuit to obtain feedback information regarding theelectrical behaviour of DEA; and a computer adapted to calculate thecapacitance of the DEA as the difference between the total instantaneouscurrent through the DEA and the product of the voltage difference acrossthe DEA and an error term, said difference being divided by the firstderivative of the voltage difference across the actuator with respect totime.
 2. The system of claim 1, wherein the DEA monitoring circuitcomprises: a pulse-width modulated current source; a dielectricelastomer actuator (DEA); a voltage sensor for measuring the voltagedifference across the DEA; a derivative sensor for measuring the firstderivative of the voltage difference across the DEA with respect totime; and a current sensor for measuring the total instantaneous currentthrough the DEA.
 3. The system of claim 2, wherein the voltage sensorcomprises a resistor ladder, the derivative sensor comprises adifferentiator circuit, and/or the current sensor comprises a seriesresistor.
 4. The system of claim 1, wherein the error term is equal tothe total instantaneous current through the DEA divided by the voltagedifference across the DEA when the first derivative of the voltagedifference across the DEA with respect to time is equal to zero.
 5. Thesystem of claim 1, wherein the computer is further adapted to determinethe state of the DEA corresponding with the calculated capacitance. 6.The system of claim 1, wherein the computer is further adapted tomonitor the leakage current of the DEA, and to limit the charge upon theelectrodes when the leakage current exceeds a predefined threshold.
 7. Amethod of determining the capacitance between opposing electrodes of adielectric elastomer actuator (DEA), the method comprising the steps of:measuring the voltage difference between the electrodes of the DEA;determining the first derivative of the voltage difference between theelectrodes of the DEA with respect to time; measuring the totalinstantaneous current through the DEA; and calculating the capacitanceof the DEA as the difference between the total instantaneous currentthrough the DEA and the product of the voltage between the electrodes ofthe DEA and an error term, said difference being divided by the firstderivative of the voltage between the electrodes of the actuator withrespect to time.
 8. The method of claim 7, wherein the error term isequal to the total instantaneous current through the DEA divided by thevoltage difference between the electrodes of the DEA when the firstderivative of the voltage difference between the electrodes of the DEAwith respect to time is equal to zero.
 9. The method of claim 7, whereinthe DEA is supplied by a pulse-width modulated (PWM) current sourcehaving a limited slew rate on either or both edges.
 10. The method ofclaim 9, wherein the slew rate of the current source is selected toenable accurate detection of the zero-crossing of the first derivativeof the voltage difference between the electrodes of the DEA with respectto time, thereby minimising sensitivity to errors in the sample timingwhen sampling the total instantaneous current through the DEA.
 11. Themethod of claim 7, wherein the step of measuring the voltage differencebetween the electrodes of the DEA comprises approximating the voltageusing a resistor ladder.
 12. The method of claim 7, wherein the step ofdetermining the first derivative of the voltage difference between theelectrodes of the DEA with respect to time comprises approximating thefirst derivative using a differentiator circuit.
 13. The method of claim7, wherein the step of determining the first derivative of the voltagedifference between the electrodes of the DEA with respect to timecomprises approximating the first derivative using a finite differencemethod on sequential measurements of the voltage difference between theelectrodes of the DEA.
 14. The method of claim 7, wherein the step ofmeasuring the total instantaneous current through the DEA comprises thestep of measuring the voltage difference across a known seriesresistance.
 15. A method of determining the leakage current betweenopposing electrodes of a dielectric elastomer actuator (DEA), the methodcomprising the steps of: determining the capacitance between theelectrodes according to the method of claim 7; calculating the firstderivative of the capacitance with respect to time; and calculating theleakage current as the difference between the total instantaneouscurrent through the DEA and the product of the voltage difference acrossthe DEA and the first derivative of the capacitance with respect totime, at a point in time when the first derivative of the voltagedifference across the DEA with respect to time is substantially equal tozero.
 16. A method of determining the state of a dielectric elastomeractuator (DEA), the method comprising the steps of: determining thecapacitance between opposing electrodes of the actuator according to themethod of claim 7; and determining the state of the DEA correspondingwith the capacitance.
 17. The method of claim 16, wherein the step ofdetermining the state of the DEA comprises the steps of: determining acapacitance ratio of the instantaneous capacitance of the DEA to itsinitial capacitance; relating the capacitance ratio to the area ratiobetween the instantaneous planar area of the DEA to its initial planararea; and determining the displacement of the DEA from the relationshipbetween the capacitance and area ratios.
 18. A method of controlling adielectric elastomer actuator (DEA) comprising the steps of: determiningthe state of the DEA according to the method of claim 16; andcontrolling the charge on the electrodes of the DEA according to thedifference between the determined state and a desired state of the DEA.19. The method of claim 18 further comprising the steps of: determininga leakage current between opposing electrodes of the DEA by obtainingthe capacitance between the opposing electrodes of the DEA; calculating,with respect to time, the first derivative of the capacitance betweenthe opposing electrodes of the DEA; calculating the leakage current asthe difference between the total instantaneous current through the DEAand the product of the voltage difference across the DEA and the firstderivative of the capacitance with respect to time, at a point in timewhen the first derivative of the voltage difference across the DEA withrespect to time is substantially equal to zero; and limiting the chargeupon the electrodes when the leakage current exceeds a predefinedthreshold.
 20. A method of determining the state of a dielectricelastomer actuator (DEA), the method comprising the steps of: a)determining the ratio between the total instantaneous current throughthe DEA and the voltage difference across the DEA at a time when thechange in voltage difference across the DEA is zero or close to zero; b)multiplying the ratio of step a) with the voltage difference across theDEA when the change in voltage is non-zero; c) subtracting the productof step b) from the total instantaneous current through the DEA; d)dividing the sum of step c) by the instantaneous change in voltagedifference across the DEA with respect to time; and e) using the productof step d) to determine the state of the DEA. 21-22. (canceled)